This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A089451 #22 Aug 18 2025 12:12:32
%S A089451 1,-1,0,1,1,0,0,0,1,0,-1,0,0,-1,1,0,1,0,-1,-1,0,-1,1,0,0,0,-1,1,0,0,0,
%T A089451 -1,0,-1,0,0,0,0,1,0,1,0,-1,0,0,0,1,-1,1,0,0,-1,0,0,0,1,0,0,0,0,-1,0,
%U A089451 0,-1,0,0,1,0,1,0,0,1,-1,0,0,1,0,0,0,0,-1,0,-1,0,-1,-1,0,0,0,1,1,1,0,0,-1,1,0,0,0,0,1,0,1,0,1,0,1,0,-1,0,-1,0,0,-1,0,0
%N A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).
%C A089451 Note that A049092 lists prime(n) such that a(n) = 0. Similarly, A078330 lists prime(n) such that a(n) = -1. See A088179 for prime(n) such that a(n) = 1. Also note that a(n) == A088144(n) (mod prime(n)).
%D A089451 J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 236.
%H A089451 Vincenzo Librandi, <a href="/A089451/b089451.txt">Table of n, a(n) for n = 1..1000</a>
%H A089451 Ed Pegg, Jr., <a href="https://www.mathpuzzle.com/MAA/02-Mobius%20Function/mathgames_11_03_03.html">Moebius Function (and squarefree numbers)</a>
%H A089451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%F A089451 a(n) = A067460(n) - 1. - _Benoit Cloitre_, Nov 04 2003
%F A089451 If p = prime(n), then a(n) is congruent modulo p to the sum of all primitive roots modulo p. [Uspensky and Heaslet]. - _Michael Somos_, Feb 16 2020
%t A089451 Table[MoebiusMu[Prime[n]-1], {n, 150}]
%o A089451 (PARI) a(n)=moebius(prime(n)-1)
%o A089451 (Magma) [MoebiusMu(NthPrime(n)-1): n in [1..100]]; // _Vincenzo Librandi_, Dec 23 2018
%Y A089451 Cf. A089495 (mu(p+1) for prime p), A089496 (mu(p+1)+mu(p-1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).
%Y A089451 Cf. A049092, A078330, A088144, A088179.
%K A089451 sign
%O A089451 1,1
%A A089451 _T. D. Noe_, Nov 03 2003